Assume $X_i$ are i.i.d, from Poisson$(\lambda)$ distribution for some $\lambda >0$, Show $\sqrt{n}(1/\bar{X}-1/\lambda)\to_D N(0,\sigma_{\lambda}^2)$

Question

Assume $X_1,...,X_n$ are i.i.d, from Poisson$(\lambda)$ distribution for some $\lambda >0$, Show that $\sigma_{\lambda}^2$ $\sqrt{n}(1/\bar{X}-1/\lambda)\to_D N(0,\sigma_{\lambda}^2)$, then compute $\sigma_{\lambda}^2$

I see this already looks similar to the statement about Central limit theorem, I thought I would solve by letting $Y_i=\frac{1}{X_i}$, however I would only know that $E(Y)\geq \frac{1}{\lambda}$

Answer 1

Note the CLT tells you $$\sqrt n (\bar X-\lambda)\rightarrow_d N(0,\lambda).$$

Now use the delta method to find the asymptotic distribution of

$$\sqrt n (g(\bar X)-g(\lambda))$$

taking $g:t\rightarrow 1/t$.

---

Update: Alternatively, write

$$\sqrt n \left(\frac{1}{\bar X}-\frac{1}{\lambda}\right)=\left(-\frac{1}{\lambda \bar X}\right)\sqrt n (\bar X-\lambda),$$

and the weak law of large numbers tells us $\bar X\rightarrow_p \lambda.$ Now use Slutsky's theorem.